Accurate inertial sensing is critical to the performance of sensing the “attitude” of working equipment (i.e. the rotation of working equipment with respect to a reference frame, usually a theoretically level ground surface). In precision agriculture, knowledge of the attitude of a vehicle is required to compensate for movements of the GNSS antenna through terrain undulation. In surveying, GNSS antennas are often mounted on a pole and to correctly determine the position of the foot of the pole, the attitude of the pole must be determined.
Inertial sensors include gyros, which measure the rate of change of angle, and accelerometers, which measure linear acceleration. Measurements from inertial sensors contain biases and other errors that must be compensated for. The measurement of an inertial sensor can be modeled by the following equation:â=Ka+bt+B(T)+ωn Where:    â is the measured inertial quantity    K is the scale factor (sensitivity) of the device    a is the true inertial quantity    bt is the stochastic bias, varying randomly with time    B(T) is the temperature-dependant bias    ωn is sensor noise, assumed to be white and Gaussian
The above equation applies equally to accelerometers and gyros, each measuring acceleration and rotation rates respectively. When the working equipment is stationary, accelerometers will read a portion of gravity, depending on the attitude of the equipment, and the gyros will read a portion of the Earth's rotation rate which is also dependant on the attitude of the equipment. When using industrial grade gyros, the contribution of the Earth's rotation rate is small enough when compared to the other error sources to be assumed to be zero to simplify the analysis without introducing significant error. With sufficient measurements at the same temperature, the contribution from the sensor noise term is small and can be incorporated into the stochastic bias. The model can then be reduced to:â= d+B(T)+εWhere:    â is the measured inertial quantity    ā=Ka is the true inertial quantity, modified by the scale factor    ε is the remaining errors, lumped into an individual term
The temperature-dependant bias is usually the dominant error. The temperature-dependant bias is not constant over temperature but varies over the operating temperature range for the inertial sensors. The temperature-dependant bias is itself not constant for a given temperature and will slowly change over time as the inertial sensor ages.
To compensate for the temperature-dependant bias, some industrial grade inertial sensors are initially calibrated to include a thermal bias error model. Due to time and cost constraints, calibration may only include actual temperature variation of the inertial senor over a limited temperature range and not a full temperature range in which the inertial senor may ultimately operate. The thermal bias error model must be updated as the inertial sensor ages. Updating the thermal bias error model is commonly done by yearly factory calibration, or calibration by means of other sensors (e.g. a multiple-GPS antenna solution). All of these strategies add cost and complexity to obtaining suitably accurate attitude solutions from the inertial sensors.
When working equipment such as a vehicle is operating, it is difficult to separate inertial sensor signal changes due to vehicle movement and vibration from a change in signal due to changes in temperature. It is therefore useful to attempt to observe the output signal of the inertial sensors while the vehicle is stationary.
U.S. Pat. Nos. 6,374,190, 6,577,952 and 5,297,028 all describe in-field auto-calibration of inertial sensors by taking a single inertial sensor and temperature sensor signal sample for each inertial sensor while their associated vehicles are stationary but operational. U.S. Pat. No. 5,527,003 describes in-field auto-calibration during the “extended alignment period that precedes taxiing of the aeroplane” and during which period gyro drift over a temperature range is sampled. The in-field auto-calibration taught by the prior art patents inherently suffer accuracy problems as the inertial sensor signals sampled include vibration errors due to the vibration caused by the vehicle's engine. The samples are also taken over a limited temperature range.